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Confluent hypergeometric function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a confluent <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a> is a solution of a confluent hypergeometric equation, which is a degenerate form of a <a href="Hypergeometric_differential_equation" class="mw-redirect" title="Hypergeometric differential equation">hypergeometric differential equation</a> where two of the three <a href="Regular_singular_point" title="Regular singular point">regular singularities</a> merge into an <a href="Irregular_singularity" class="mw-redirect" title="Irregular singularity">irregular singularity</a>. The term confluent refers to the merging of singular points of families of differential equations; confluere is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions:

<ul><li>Kummer's (confluent hypergeometric) function M(a, b, z), introduced by <a href="Ernst_Kummer" title="Ernst Kummer">Kummer</a> (<a href="#CITEREFKummer1837">1837</a>), is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated <a href="Kummer's_function" title="Kummer's function">Kummer's function</a> bearing the same name.</li>

<li>Tricomi's (confluent hypergeometric) function U(a, b, z) introduced by <a href="Francesco_Tricomi" title="Francesco Tricomi">Francesco Tricomi</a> (<a href="#CITEREFTricomi1947">1947</a>), sometimes denoted by Ψ(a; b; z), is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind.</li>

<li><a href="Whittaker_function" title="Whittaker function">Whittaker functions</a> (for <a href="Edmund_Taylor_Whittaker" class="mw-redirect" title="Edmund Taylor Whittaker">Edmund Taylor Whittaker</a>) are solutions to Whittaker's equation.</li>

<li><a href="Coulomb_wave_function" title="Coulomb wave function">Coulomb wave functions</a> are solutions to the Coulomb wave equation.</li></ul>

The Kummer functions, Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables.

<div class="mw-heading mw-heading2"><h2 id="Kummer's_equation">Kummer's equation</h2></div>

Kummer's equation may be written as:

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<annotation encoding="application/x-tex">{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0,}</annotation>

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</math><img src="./3cc490560235da68e9175330b19b8220bf839be3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.795ex; height:6.009ex;" alt="{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0,}" loading="lazy"></dd></dl>

with a regular singular point at z = 0 and an irregular singular point at z = ∞. It has two (usually) <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> solutions M(a, b, z) and U(a, b, z).

Kummer's function of the first kind M is a <a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">generalized hypergeometric series</a> introduced in (<a href="#CITEREFKummer1837">Kummer 1837</a>), given by:

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<annotation encoding="application/x-tex">{\displaystyle M(a,b,z)=\sum _{n=0}^{\infty }{\frac {a^{(n)}z^{n}}{b^{(n)}n!}}={}_{1}F_{1}(a;b;z),}</annotation>

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</math><img src="./19bbed87ba6981fe69165d77f4608d0ee3280a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.889ex; height:6.843ex;" alt="{\displaystyle M(a,b,z)=\sum _{n=0}^{\infty }{\frac {a^{(n)}z^{n}}{b^{(n)}n!}}={}_{1}F_{1}(a;b;z),}" loading="lazy"></dd></dl>

where:

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<annotation encoding="application/x-tex">{\displaystyle a^{(0)}=1,}</annotation>

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</math><img src="./b785bc8ac069bf34955a01eb1ffc96d04672b1b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.471ex; height:3.176ex;" alt="{\displaystyle a^{(0)}=1,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle a^{(n)}=a(a+1)(a+2)\cdots (a+n-1)\,,}</annotation>

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</math><img src="./bbce7e5cf63769b6b8beeb126abf509e1db85076.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.948ex; height:3.343ex;" alt="{\displaystyle a^{(n)}=a(a+1)(a+2)\cdots (a+n-1)\,,}" loading="lazy"></dd></dl>

is the <a href="Rising_factorial" class="mw-redirect" title="Rising factorial">rising factorial</a>. Another common notation for this solution is Φ(a, b, z). Considered as a function of a, b, or z with the other two held constant, this defines an <a href="Entire_function" title="Entire function">entire function</a> of a or z, except when b = 0, −1, −2, ... As a function of b it is <a href="Analytic_function" title="Analytic function">analytic</a> except for poles at the non-positive integers.

Some values of a and b yield solutions that can be expressed in terms of other known functions. See <a href="#Special_cases">#Special cases</a>. When a is a non-positive integer, then Kummer's function (if it is defined) is a generalized <a href="Laguerre_polynomial" class="mw-redirect" title="Laguerre polynomial">Laguerre polynomial</a>.

Just as the confluent differential equation is a limit of the <a href="Hypergeometric_differential_equation" class="mw-redirect" title="Hypergeometric differential equation">hypergeometric differential equation</a> as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of the <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a>

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<annotation encoding="application/x-tex">{\displaystyle M(a,c,z)=\lim _{b\to \infty }{}_{2}F_{1}(a,b;c;z/b)}</annotation>

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</math><img src="./b2aac32fabc9be3b7f58d8dcd5b8de46608c7d7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.118ex; height:4.009ex;" alt="{\displaystyle M(a,c,z)=\lim _{b\to \infty }{}_{2}F_{1}(a,b;c;z/b)}" loading="lazy"></dd></dl>

and many of the properties of the confluent hypergeometric function are limiting cases of properties of the hypergeometric function.

Since Kummer's equation is second order there must be another, independent, solution. The <a href="Indicial_equation" class="mw-redirect" title="Indicial equation">indicial equation</a> of the method of Frobenius tells us that the lowest power of a <a href="Power_series" title="Power series">power series</a> solution to the Kummer equation is either 0 or 1 − b. If we let w(z) be

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<annotation encoding="application/x-tex">{\displaystyle w(z)=z^{1-b}v(z)}</annotation>

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</math><img src="./52096de7435a1cd1423560d8966ef80e81fdc6d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.814ex; height:3.176ex;" alt="{\displaystyle w(z)=z^{1-b}v(z)}" loading="lazy"></dd></dl>

then the differential equation gives

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<annotation encoding="application/x-tex">{\displaystyle z^{2-b}{\frac {d^{2}v}{dz^{2}}}+2(1-b)z^{1-b}{\frac {dv}{dz}}-b(1-b)z^{-b}v+(b-z)\left[z^{1-b}{\frac {dv}{dz}}+(1-b)z^{-b}v\right]-az^{1-b}v=0}</annotation>

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</math><img src="./452828e42ddeec686233652c5f0b08e758fd1f36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:88.964ex; height:6.343ex;" alt="{\displaystyle z^{2-b}{\frac {d^{2}v}{dz^{2}}}+2(1-b)z^{1-b}{\frac {dv}{dz}}-b(1-b)z^{-b}v+(b-z)\left[z^{1-b}{\frac {dv}{dz}}+(1-b)z^{-b}v\right]-az^{1-b}v=0}" loading="lazy"></dd></dl>

which, upon dividing out z1−b and simplifying, becomes

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<annotation encoding="application/x-tex">{\displaystyle z{\frac {d^{2}v}{dz^{2}}}+(2-b-z){\frac {dv}{dz}}-(a+1-b)v=0.}</annotation>

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</math><img src="./e329e624290b3c9fbdf6816f2abf32db70d8cb6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.838ex; height:6.009ex;" alt="{\displaystyle z{\frac {d^{2}v}{dz^{2}}}+(2-b-z){\frac {dv}{dz}}-(a+1-b)v=0.}" loading="lazy"></dd></dl>

This means that z1−bM(a + 1 − b, 2 − b, z) is a solution so long as b is not an integer greater than 1, just as M(a, b, z) is a solution so long as b is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function U(a, b, z) introduced by <a href="Francesco_Tricomi" title="Francesco Tricomi">Francesco Tricomi</a> (<a href="#CITEREFTricomi1947">1947</a>), and sometimes denoted by Ψ(a; b; z). It is a combination of the above two solutions, defined by

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<annotation encoding="application/x-tex">{\displaystyle U(a,b,z)={\frac {\Gamma (1-b)}{\Gamma (a+1-b)}}M(a,b,z)+{\frac {\Gamma (b-1)}{\Gamma (a)}}z^{1-b}M(a+1-b,2-b,z).}</annotation>

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</math><img src="./ade28eae1a189caa562db8ab30ec03cd2d0e016b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:73.07ex; height:6.509ex;" alt="{\displaystyle U(a,b,z)={\frac {\Gamma (1-b)}{\Gamma (a+1-b)}}M(a,b,z)+{\frac {\Gamma (b-1)}{\Gamma (a)}}z^{1-b}M(a+1-b,2-b,z).}" loading="lazy"></dd></dl>

Although this expression is undefined for integer b, it has the advantage that it can be extended to any integer b by continuity. Unlike Kummer's function which is an <a href="Entire_function" title="Entire function">entire function</a> of z, U(z) usually has a <a href="Singularity_(mathematics)" title="Singularity (mathematics)">singularity</a> at zero. For example, if b = 0 and a ≠ 0 then Γ(a+1)U(a, b, z) − 1 is asymptotic to az ln z as z goes to zero. But see <a href="#Special_cases">#Special cases</a> for some examples where it is an entire function (polynomial).

Note that the solution z1−bU(a + 1 − b, 2 − b, z) to Kummer's equation is the same as the solution U(a, b, z), see <a href="#Kummer's_transformation">#Kummer's transformation</a>.

For most combinations of real or complex a and b, the functions M(a, b, z) and U(a, b, z) are independent, and if b is a non-positive integer, so M(a, b, z) doesn't exist, then we may be able to use z1−bM(a+1−b, 2−b, z) as a second solution. But if a is a non-positive integer and b is not a non-positive integer, then U(z) is a multiple of M(z). In that case as well, z1−bM(a+1−b, 2−b, z) can be used as a second solution if it exists and is different. But when b is an integer greater than 1, this solution doesn't exist, and if b = 1 then it exists but is a multiple of U(a, b, z) and of M(a, b, z) In those cases a second solution exists of the following form and is valid for any real or complex a and any positive integer b except when a is a positive integer less than b:

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<annotation encoding="application/x-tex">{\displaystyle M(a,b,z)\ln z+z^{1-b}\sum _{k=0}^{\infty }C_{k}z^{k}}</annotation>

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</math><img src="./aab23228c66044ff65eb22419f2f7324b98c68e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.464ex; height:7.009ex;" alt="{\displaystyle M(a,b,z)\ln z+z^{1-b}\sum _{k=0}^{\infty }C_{k}z^{k}}" loading="lazy"></dd></dl>

When a = 0 we can alternatively use:

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{z}(-u)^{-b}e^{u}\mathrm {d} u.}</annotation>

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</math><img src="./e985329b5c34bfb059a48fc94e19ebab575feefc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.135ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{z}(-u)^{-b}e^{u}\mathrm {d} u.}" loading="lazy"></dd></dl>

When b = 1 this is the <a href="Exponential_integral" title="Exponential integral">exponential integral</a> E1(−z).

A similar problem occurs when a−b is a negative integer and b is an integer less than 1. In this case M(a, b, z) doesn't exist, and U(a, b, z) is a multiple of z1−bM(a+1−b, 2−b, z). A second solution is then of the form:

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<annotation encoding="application/x-tex">{\displaystyle z^{1-b}M(a+1-b,2-b,z)\ln z+\sum _{k=0}^{\infty }C_{k}z^{k}}</annotation>

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</math><img src="./1727ad0321e8c83c7a97e5389b435087a4a029f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.921ex; height:7.009ex;" alt="{\displaystyle z^{1-b}M(a+1-b,2-b,z)\ln z+\sum _{k=0}^{\infty }C_{k}z^{k}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Other_equations">Other equations</h3></div>

Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as:

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<annotation encoding="application/x-tex">{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-\left(\sum _{m=0}^{M}a_{m}z^{m}\right)w=0}</annotation>

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</math><img src="./280109ffce59d02ba095dff14d1568ac6bc5bd4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.587ex; height:7.509ex;" alt="{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-\left(\sum _{m=0}^{M}a_{m}z^{m}\right)w=0}" loading="lazy"> <a href="#cite_note-1">[1]</a></dd></dl>

Note that for M = 0 or when the summation involves just one term, it reduces to the conventional Confluent Hypergeometric Equation.

Thus Confluent Hypergeometric Functions can be used to solve "most" second-order ordinary differential equations whose variable coefficients are all linear functions of z, because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation:

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<annotation encoding="application/x-tex">{\displaystyle (A+Bz){\frac {d^{2}w}{dz^{2}}}+(C+Dz){\frac {dw}{dz}}+(E+Fz)w=0}</annotation>

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</math><img src="./51d78bdcdc0febb2531dece35c85bae161c186de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:48.022ex; height:6.009ex;" alt="{\displaystyle (A+Bz){\frac {d^{2}w}{dz^{2}}}+(C+Dz){\frac {dw}{dz}}+(E+Fz)w=0}" loading="lazy"></dd></dl>

First we move the <a href="Regular_singular_point" title="Regular singular point">regular singular point</a> to 0 by using the substitution of A + Bz ↦ z, which converts the equation to:

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<annotation encoding="application/x-tex">{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(C+Dz){\frac {dw}{dz}}+(E+Fz)w=0}</annotation>

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</math><img src="./5188dd0097cc3b8a48b71a5bcb29f14a14bc00e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:39.865ex; height:6.009ex;" alt="{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(C+Dz){\frac {dw}{dz}}+(E+Fz)w=0}" loading="lazy"></dd></dl>

with new values of C, D, E, and F. Next we use the substitution:

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<annotation encoding="application/x-tex">{\displaystyle z\mapsto {\frac {1}{\sqrt {D^{2}-4F}}}z}</annotation>

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</math><img src="./3e82976cc97f6ee6aee0def1cb81480f7174b319.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.672ex; height:6.509ex;" alt="{\displaystyle z\mapsto {\frac {1}{\sqrt {D^{2}-4F}}}z}" loading="lazy"></dd></dl>

and multiply the equation by the same factor, obtaining:

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<annotation encoding="application/x-tex">{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+\left(C+{\frac {D}{\sqrt {D^{2}-4F}}}z\right){\frac {dw}{dz}}+\left({\frac {E}{\sqrt {D^{2}-4F}}}+{\frac {F}{D^{2}-4F}}z\right)w=0}</annotation>

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</math><img src="./f9a40a800bdd22eb86d819d545d464de9c90fd3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:72.265ex; height:7.509ex;" alt="{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+\left(C+{\frac {D}{\sqrt {D^{2}-4F}}}z\right){\frac {dw}{dz}}+\left({\frac {E}{\sqrt {D^{2}-4F}}}+{\frac {F}{D^{2}-4F}}z\right)w=0}" loading="lazy"></dd></dl>

whose solution is

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(-\left(1+{\frac {D}{\sqrt {D^{2}-4F}}}\right){\frac {z}{2}}\right)w(z),}</annotation>

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</math><img src="./a9e9db07e71325c3d2250425a2731f4704b13135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.589ex; height:7.509ex;" alt="{\displaystyle \exp \left(-\left(1+{\frac {D}{\sqrt {D^{2}-4F}}}\right){\frac {z}{2}}\right)w(z),}" loading="lazy"></dd></dl>

where w(z) is a solution to Kummer's equation with

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<annotation encoding="application/x-tex">{\displaystyle a=\left(1+{\frac {D}{\sqrt {D^{2}-4F}}}\right){\frac {C}{2}}-{\frac {E}{\sqrt {D^{2}-4F}}},\qquad b=C.}</annotation>

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</math><img src="./74691fbcb4e605525052a18b1312109667b0ac36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.795ex; height:7.509ex;" alt="{\displaystyle a=\left(1+{\frac {D}{\sqrt {D^{2}-4F}}}\right){\frac {C}{2}}-{\frac {E}{\sqrt {D^{2}-4F}}},\qquad b=C.}" loading="lazy"></dd></dl>

Note that the square root may give an imaginary or complex number. If it is zero, another solution must be used, namely

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(-{\tfrac {1}{2}}Dz\right)w(z),}</annotation>

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</math><img src="./660f7734957b9107cd4a32cba5a813b286471762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.369ex; height:3.509ex;" alt="{\displaystyle \exp \left(-{\tfrac {1}{2}}Dz\right)w(z),}" loading="lazy"></dd></dl>

where w(z) is a <a href="Confluent_hypergeometric_limit_function" class="mw-redirect" title="Confluent hypergeometric limit function">confluent hypergeometric limit function</a> satisfying

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<annotation encoding="application/x-tex">{\displaystyle zw''(z)+Cw'(z)+\left(E-{\tfrac {1}{2}}CD\right)w(z)=0.}</annotation>

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</math><img src="./b839fb0e104d2651a0e4a82ef057dcd5a63d7e2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.431ex; height:3.509ex;" alt="{\displaystyle zw''(z)+Cw'(z)+\left(E-{\tfrac {1}{2}}CD\right)w(z)=0.}" loading="lazy"></dd></dl>

As noted below, even the <a href="Bessel_equation" class="mw-redirect" title="Bessel equation">Bessel equation</a> can be solved using confluent hypergeometric functions.

<div class="mw-heading mw-heading2"><h2 id="Integral_representations">Integral representations</h2></div>

If Re b > Re a > 0, M(a, b, z) can be represented as an integral

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<annotation encoding="application/x-tex">{\displaystyle M(a,b,z)={\frac {\Gamma (b)}{\Gamma (a)\Gamma (b-a)}}\int _{0}^{1}e^{zu}u^{a-1}(1-u)^{b-a-1}\,du.}</annotation>

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</math><img src="./c6ef5b1a1bc60ec59d817302d99de8cbe376d183.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.149ex; height:6.509ex;" alt="{\displaystyle M(a,b,z)={\frac {\Gamma (b)}{\Gamma (a)\Gamma (b-a)}}\int _{0}^{1}e^{zu}u^{a-1}(1-u)^{b-a-1}\,du.}" loading="lazy"></dd></dl>

thus M(a, a+b, it) is the <a href="Characteristic_function_(probability)" class="mw-redirect" title="Characteristic function (probability)">characteristic function</a> of the <a href="Beta_distribution" title="Beta distribution">beta distribution</a>. For a with positive real part U can be obtained by the <a href="Laplace_transform" title="Laplace transform">Laplace integral</a>

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<annotation encoding="application/x-tex">{\displaystyle U(a,b,z)={\frac {1}{\Gamma (a)}}\int _{0}^{\infty }e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt,\quad (\operatorname {Re} \ a>0)}</annotation>

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</math><img src="./b16f4524cff26ddd4c6d9b6a03f5961d0dc94126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.163ex; height:6.176ex;" alt="{\displaystyle U(a,b,z)={\frac {1}{\Gamma (a)}}\int _{0}^{\infty }e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt,\quad (\operatorname {Re} \ a>0)}" loading="lazy"></dd></dl>

The integral defines a solution in the right half-plane Re z > 0.

They can also be represented as <a href="Barnes_integral" title="Barnes integral">Barnes integrals</a>

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<annotation encoding="application/x-tex">{\displaystyle M(a,b,z)={\frac {1}{2\pi i}}{\frac {\Gamma (b)}{\Gamma (a)}}\int _{-i\infty }^{i\infty }{\frac {\Gamma (-s)\Gamma (a+s)}{\Gamma (b+s)}}(-z)^{s}ds}</annotation>

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</math><img src="./6e9c9698271e1c5abfb46a24079eb5da099097b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.417ex; height:6.509ex;" alt="{\displaystyle M(a,b,z)={\frac {1}{2\pi i}}{\frac {\Gamma (b)}{\Gamma (a)}}\int _{-i\infty }^{i\infty }{\frac {\Gamma (-s)\Gamma (a+s)}{\Gamma (b+s)}}(-z)^{s}ds}" loading="lazy"></dd></dl>

where the contour passes to one side of the poles of Γ(−s) and to the other side of the poles of Γ(a + s).

<div class="mw-heading mw-heading2"><h2 id="Asymptotic_behavior">Asymptotic behavior</h2></div>

If a solution to Kummer's equation is asymptotic to a power of z as z → ∞, then the power must be −a. This is in fact the case for Tricomi's solution U(a, b, z). Its <a href="Asymptotic" class="mw-redirect" title="Asymptotic">asymptotic</a> behavior as z → ∞ can be deduced from the integral representations. If z = x ∈ R, then making a change of variables in the integral followed by expanding the <a href="Binomial_series" title="Binomial series">binomial series</a> and integrating it formally term by term gives rise to an <a href="Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a> expansion, valid as x → ∞:<a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle U(a,b,x)\sim x^{-a}\,_{2}F_{0}\left(a,a-b+1;\,;-{\frac {1}{x}}\right),}</annotation>

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</math><img src="./6d47e86981d999a5858e8a52055620b0ebf52a02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.621ex; height:6.176ex;" alt="{\displaystyle U(a,b,x)\sim x^{-a}\,_{2}F_{0}\left(a,a-b+1;\,;-{\frac {1}{x}}\right),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{2}F_{0}(\cdot ,\cdot ;;-1/x)}">

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<annotation encoding="application/x-tex">{\displaystyle _{2}F_{0}(\cdot ,\cdot ;;-1/x)}</annotation>

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</math><img src="./373defd06d91891169d3fec265e2ce8bf200b7e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.271ex; height:2.843ex;" alt="{\displaystyle _{2}F_{0}(\cdot ,\cdot ;;-1/x)}" loading="lazy"> is a <a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">generalized hypergeometric series</a> with 1 as leading term, which generally converges nowhere, but exists as a <a href="Formal_power_series" title="Formal power series">formal power series</a> in 1/x. This <a href="Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a> is also valid for complex z instead of real x, with |arg z| < 3π/2.

The asymptotic behavior of Kummer's solution for large |z| is:

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<annotation encoding="application/x-tex">{\displaystyle M(a,b,z)\sim \Gamma (b)\left({\frac {e^{z}z^{a-b}}{\Gamma (a)}}+{\frac {(-z)^{-a}}{\Gamma (b-a)}}\right)}</annotation>

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</math><img src="./ead7e3ea6f3e831ef27919cf8b16204e2b62cde2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.905ex; height:6.509ex;" alt="{\displaystyle M(a,b,z)\sim \Gamma (b)\left({\frac {e^{z}z^{a-b}}{\Gamma (a)}}+{\frac {(-z)^{-a}}{\Gamma (b-a)}}\right)}" loading="lazy"></dd></dl>

The powers of z are taken using −3π/2 < arg z ≤ π/2.<a href="#cite_note-3">[3]</a> The first term is not needed when Γ(b − a) is finite, that is when b − a is not a non-positive integer and the real part of z goes to negative infinity, whereas the second term is not needed when Γ(a) is finite, that is, when a is a not a non-positive integer and the real part of z goes to positive infinity.

There is always some solution to Kummer's equation asymptotic to ezza−b as z → −∞. Usually this will be a combination of both M(a, b, z) and U(a, b, z) but can also be expressed as ez (−1)a-b U(b − a, b, −z).

<div class="mw-heading mw-heading2"><h2 id="Relations">Relations</h2></div>

There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples.

<div class="mw-heading mw-heading3"><h3 id="Contiguous_relations">Contiguous relations</h3></div>

Given M(a, b, z), the four functions M(a ± 1, b, z), M(a, b ± 1, z) are called contiguous to M(a, b, z). The function M(a, b, z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a, b, and z. This gives (4 2) = 6 relations, given by identifying any two lines on the right hand side of

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}z{\frac {dM}{dz}}=z{\frac {a}{b}}M(a+,b+)&=a(M(a+)-M)\\&=(b-1)(M(b-)-M)\\&=(b-a)M(a-)+(a-b+z)M\\&=z(a-b)M(b+)/b+zM\\\end{aligned}}}</annotation>

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</math><img src="./060cde4ab1b7950c53803b2a2eb96c1033644fe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.562ex; margin-bottom: -0.276ex; width:57.069ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}z{\frac {dM}{dz}}=z{\frac {a}{b}}M(a+,b+)&=a(M(a+)-M)\\&=(b-1)(M(b-)-M)\\&=(b-a)M(a-)+(a-b+z)M\\&=z(a-b)M(b+)/b+zM\\\end{aligned}}}" loading="lazy"></dd></dl>

In the notation above, M = M(a, b, z), M(a+) = M(a + 1, b, z), and so on.

Repeatedly applying these relations gives a linear relation between any three functions of the form M(a + m, b + n, z) (and their higher derivatives), where m, n are integers.

There are similar relations for U.

<div class="mw-heading mw-heading3"><h3 id="Kummer's_transformation">Kummer's transformation</h3></div>

Kummer's functions are also related by Kummer's transformations:

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<annotation encoding="application/x-tex">{\displaystyle M(a,b,z)=e^{z}\,M(b-a,b,-z)}</annotation>

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</math><img src="./b82fb0265eb98a2fcb44c15152ded4e5ba4d20ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.487ex; height:2.843ex;" alt="{\displaystyle M(a,b,z)=e^{z}\,M(b-a,b,-z)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle U(a,b,z)=z^{1-b}U\left(1+a-b,2-b,z\right)}</annotation>

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</math><img src="./1ed96ccd8f7731edf923b091f18881a758ba41d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.408ex; height:3.176ex;" alt="{\displaystyle U(a,b,z)=z^{1-b}U\left(1+a-b,2-b,z\right)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Multiplication_theorem">Multiplication theorem</h2></div>

The following <a href="Multiplication_theorem" title="Multiplication theorem">multiplication theorems</a> hold true:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(a,b,z)&=e^{(1-t)z}\sum _{i=0}{\frac {(t-1)^{i}z^{i}}{i!}}U(a,b+i,zt)\\&=e^{(1-t)z}t^{b-1}\sum _{i=0}{\frac {\left(1-{\frac {1}{t}}\right)^{i}}{i!}}U(a-i,b-i,zt).\end{aligned}}}</annotation>

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</math><img src="./0e41d757c59cee3d371d3a15bf350aa38ebfad9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:54.065ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}U(a,b,z)&=e^{(1-t)z}\sum _{i=0}{\frac {(t-1)^{i}z^{i}}{i!}}U(a,b+i,zt)\\&=e^{(1-t)z}t^{b-1}\sum _{i=0}{\frac {\left(1-{\frac {1}{t}}\right)^{i}}{i!}}U(a-i,b-i,zt).\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Connection_with_Laguerre_polynomials_and_similar_representations">Connection with Laguerre polynomials and similar representations</h2></div>

In terms of <a href="Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a>, Kummer's functions have several expansions, for example

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<annotation encoding="application/x-tex">{\displaystyle M\left(a,b,{\frac {xy}{x-1}}\right)=(1-x)^{a}\cdot \sum _{n}{\frac {a^{(n)}}{b^{(n)}}}L_{n}^{(b-1)}(y)x^{n}}</annotation>

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</math><img src="./b135f334f039f05057673f28ba322768700f2846.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.455ex; height:6.843ex;" alt="{\displaystyle M\left(a,b,{\frac {xy}{x-1}}\right)=(1-x)^{a}\cdot \sum _{n}{\frac {a^{(n)}}{b^{(n)}}}L_{n}^{(b-1)}(y)x^{n}}" loading="lazy"> (<a href="#CITEREFErdélyiMagnusOberhettingerTricomi1953">Erdélyi et al. 1953</a>, 6.12)</dd></dl>

or

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<annotation encoding="application/x-tex">{\displaystyle M\left(a,\,b,\,z\right)={\frac {\Gamma \left(1-a\right)\cdot \Gamma \left(b\right)}{\Gamma \left(b-a\right)}}\cdot L_{-a}^{(b-1)}\left(z\right)}</annotation>

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</math><img src="./3baa4e5a4537424095dd0ecc1b6530a6674b6610.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.802ex; height:6.509ex;" alt="{\displaystyle M\left(a,\,b,\,z\right)={\frac {\Gamma \left(1-a\right)\cdot \Gamma \left(b\right)}{\Gamma \left(b-a\right)}}\cdot L_{-a}^{(b-1)}\left(z\right)}" loading="lazy"><a rel="nofollow" class="external autonumber" href="https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/27/01/0001/">[1]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2></div>

Functions that can be expressed as special cases of the confluent hypergeometric function include:

<ul><li>Some <a href="Elementary_function" title="Elementary function">elementary functions</a> where the left-hand side is not defined when b is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle M(0,b,z)=1}</annotation>

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</math><img src="./412f79dade34992b096554c6da22be6f9e06bce7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.828ex; height:2.843ex;" alt="{\displaystyle M(0,b,z)=1}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle U(0,c,z)=1}</annotation>

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</math><img src="./27892e37c08b927333e694599cad6c6bd2dad7e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.178ex; height:2.843ex;" alt="{\displaystyle U(0,c,z)=1}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle M(b,b,z)=e^{z}}</annotation>

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</math><img src="./430480c4025c560cf714a026424f186c0075918f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.586ex; height:2.843ex;" alt="{\displaystyle M(b,b,z)=e^{z}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle U(a,a,z)=e^{z}\int _{z}^{\infty }u^{-a}e^{-u}du}</annotation>

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</math><img src="./cc7687b7d0ed03f322997747e105028d503b8fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.295ex; height:5.843ex;" alt="{\displaystyle U(a,a,z)=e^{z}\int _{z}^{\infty }u^{-a}e^{-u}du}" loading="lazy"> (a polynomial if a is a non-positive integer)</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {U(1,b,z)}{\Gamma (b-1)}}+{\frac {M(1,b,z)}{\Gamma (b)}}=z^{1-b}e^{z}}</annotation>

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</math><img src="./5a49a0d42008a5c4106dc27d11171064ef580fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.3ex; height:6.509ex;" alt="{\displaystyle {\frac {U(1,b,z)}{\Gamma (b-1)}}+{\frac {M(1,b,z)}{\Gamma (b)}}=z^{1-b}e^{z}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle M(n,b,z)}</annotation>

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</math><img src="./af14fda4924f5129fc1ebbcfa31a75d7b00f2102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.8ex; height:2.843ex;" alt="{\displaystyle M(n,b,z)}" loading="lazy"> for non-positive integer n is a <a href="Generalized_Laguerre_polynomial" class="mw-redirect" title="Generalized Laguerre polynomial">generalized Laguerre polynomial</a>.</dd>

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<annotation encoding="application/x-tex">{\displaystyle U(n,c,z)}</annotation>

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</math><img src="./84e27b7b801f0d7a4c3dc2897a33118523de44d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.149ex; height:2.843ex;" alt="{\displaystyle U(n,c,z)}" loading="lazy"> for non-positive integer n is a multiple of a generalized Laguerre polynomial, equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\Gamma (1-c)}{\Gamma (n+1-c)}}M(n,c,z)}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\Gamma (1-c)}{\Gamma (n+1-c)}}M(n,c,z)}</annotation>

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</math><img src="./56e5d90412e59f1db4f05cc9f9163833d8d95121.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.029ex; height:4.843ex;" alt="{\displaystyle {\tfrac {\Gamma (1-c)}{\Gamma (n+1-c)}}M(n,c,z)}" loading="lazy"> when the latter exists.</dd>

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<annotation encoding="application/x-tex">{\displaystyle U(c-n,c,z)}</annotation>

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</math><img src="./0fb6aac9e0a14fadc8e1f8952ea45bb17b1dfd76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.997ex; height:2.843ex;" alt="{\displaystyle U(c-n,c,z)}" loading="lazy"> when n is a positive integer is a closed form with powers of z, equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\Gamma (c-1)}{\Gamma (c-n)}}z^{1-c}M(1-n,2-c,z)}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\Gamma (c-1)}{\Gamma (c-n)}}z^{1-c}M(1-n,2-c,z)}</annotation>

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</math><img src="./4b15f62392d53133cb90812dbe73256377595bb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.069ex; height:4.843ex;" alt="{\displaystyle {\tfrac {\Gamma (c-1)}{\Gamma (c-n)}}z^{1-c}M(1-n,2-c,z)}" loading="lazy"> when the latter exists.</dd>

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<annotation encoding="application/x-tex">{\displaystyle U(a,a+1,z)=z^{-a}}</annotation>

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</math><img src="./e97b606a70f68d3db4abf0810e07c0b719eafed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.78ex; height:3.009ex;" alt="{\displaystyle U(a,a+1,z)=z^{-a}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle U(-n,-2n,z)}</annotation>

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</math><img src="./41c1957781f943f62227185729cf5fb123298544.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.316ex; height:2.843ex;" alt="{\displaystyle U(-n,-2n,z)}" loading="lazy"> for non-negative integer n is a Bessel polynomial (see lower down).</dd>

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<annotation encoding="application/x-tex">{\displaystyle M(1,2,z)=(e^{z}-1)/z,\ \ M(1,3,z)=2!(e^{z}-1-z)/z^{2}}</annotation>

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</math><img src="./fed8215b7b21b21e09ad5ad3a04e5f966c0e4b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.947ex; height:3.176ex;" alt="{\displaystyle M(1,2,z)=(e^{z}-1)/z,\ \ M(1,3,z)=2!(e^{z}-1-z)/z^{2}}" loading="lazy"> etc.</dd>

<dd>Using the contiguous relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aM(a+)=(a+z)M+z(a-b)M(b+)/b}">

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<annotation encoding="application/x-tex">{\displaystyle aM(a+)=(a+z)M+z(a-b)M(b+)/b}</annotation>

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</math><img src="./e5bdc4aac8a67ea35695a397d2e564defa0e826e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.05ex; height:2.843ex;" alt="{\displaystyle aM(a+)=(a+z)M+z(a-b)M(b+)/b}" loading="lazy"> we get, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(2,1,z)=(1+z)e^{z}.}">

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<annotation encoding="application/x-tex">{\displaystyle M(2,1,z)=(1+z)e^{z}.}</annotation>

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</math><img src="./a6232aad3076a437176076a48886be96b2ceca00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.463ex; height:2.843ex;" alt="{\displaystyle M(2,1,z)=(1+z)e^{z}.}" loading="lazy"></dd></dl></dd></dl>

<ul><li><a href="Bateman's_function" class="mw-redirect" title="Bateman's function">Bateman's function</a></li>

<li><a href="Bessel_function" title="Bessel function">Bessel functions</a> and many related functions such as <a href="Airy_function" title="Airy function">Airy functions</a>, <a href="Kelvin_function" class="mw-redirect" title="Kelvin function">Kelvin functions</a>, <a href="Hankel_function" class="mw-redirect" title="Hankel function">Hankel functions</a>. For example, in the special case b = 2a the function reduces to a <a href="Bessel_function" title="Bessel function">Bessel function</a>:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{1}(a,2a,x)=e^{x/2}\,{}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)=e^{x/2}\left({\tfrac {x}{4}}\right)^{1/2-a}\Gamma \left(a+{\tfrac {1}{2}}\right)I_{a-1/2}\left({\tfrac {x}{2}}\right).}</annotation>

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</math><img src="./7a08877829e78464326ee0845b2eb3015acce6e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:74.666ex; height:5.343ex;" alt="{\displaystyle {}_{1}F_{1}(a,2a,x)=e^{x/2}\,{}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)=e^{x/2}\left({\tfrac {x}{4}}\right)^{1/2-a}\Gamma \left(a+{\tfrac {1}{2}}\right)I_{a-1/2}\left({\tfrac {x}{2}}\right).}" loading="lazy"></dd></dl></dd>

<dd>This identity is sometimes also referred to as <a href="Ernst_Kummer" title="Ernst Kummer">Kummer's</a> second transformation. Similarly

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<annotation encoding="application/x-tex">{\displaystyle U(a,2a,x)={\frac {e^{x/2}}{\sqrt {\pi }}}x^{1/2-a}K_{a-1/2}(x/2),}</annotation>

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</math><img src="./82368be866a3fcf24839332434da296c69a07e10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.552ex; height:6.676ex;" alt="{\displaystyle U(a,2a,x)={\frac {e^{x/2}}{\sqrt {\pi }}}x^{1/2-a}K_{a-1/2}(x/2),}" loading="lazy"></dd></dl></dd>

<dd>When a is a non-positive integer, this equals 2−aθ−a(x/2) where θ is a <a href="Bessel_polynomial" class="mw-redirect" title="Bessel polynomial">Bessel polynomial</a>.</dd></dl>

<ul><li>The <a href="Error_function" title="Error function">error function</a> can be expressed as</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt={\frac {2x}{\sqrt {\pi }}}\ {}_{1}F_{1}\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).}</annotation>

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</math><img src="./60bd2494c31656bfc1b454853ea627e8a3bab6a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:48.795ex; height:6.343ex;" alt="{\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt={\frac {2x}{\sqrt {\pi }}}\ {}_{1}F_{1}\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).}" loading="lazy"></dd></dl></dd></dl>

<ul><li><a href="Coulomb_wave_function" title="Coulomb wave function">Coulomb wave function</a></li>

<li><a href="Cunningham_function" title="Cunningham function">Cunningham functions</a></li>

<li><a href="Exponential_integral" title="Exponential integral">Exponential integral</a> and related functions such as the <a href="Sine_integral" class="mw-redirect" title="Sine integral">sine integral</a>, <a href="Logarithmic_integral" class="mw-redirect" title="Logarithmic integral">logarithmic integral</a></li>

<li><a href="Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a></li>

<li><a href="Incomplete_gamma_function" title="Incomplete gamma function">Incomplete gamma function</a></li>

<li><a href="Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a></li>

<li><a href="Parabolic_cylinder_function" title="Parabolic cylinder function">Parabolic cylinder function</a> (or Weber function)</li>

<li><a href="Poisson%E2%80%93Charlier_function" class="mw-redirect" title="Poisson–Charlier function">Poisson–Charlier function</a></li>

<li><a href="Toronto_function" title="Toronto function">Toronto functions</a></li>

<li><a href="Whittaker_function" title="Whittaker function">Whittaker functions</a> Mκ,μ(z), Wκ,μ(z) are solutions of <a href="Whittaker's_equation" class="mw-redirect" title="Whittaker's equation">Whittaker's equation</a> that can be expressed in terms of Kummer functions M and U by</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle M_{\kappa ,\mu }(z)=e^{-{\tfrac {z}{2}}}z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ;z\right)}</annotation>

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</math><img src="./e8cdfb751f073e1c4f5ba5c4503b03e264fb3418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.142ex; height:4.843ex;" alt="{\displaystyle M_{\kappa ,\mu }(z)=e^{-{\tfrac {z}{2}}}z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ;z\right)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle W_{\kappa ,\mu }(z)=e^{-{\tfrac {z}{2}}}z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ;z\right)}</annotation>

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</math><img src="./582dce67bafdf026f1a2d6584f2384d542c5c0b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.422ex; height:4.843ex;" alt="{\displaystyle W_{\kappa ,\mu }(z)=e^{-{\tfrac {z}{2}}}z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ;z\right)}" loading="lazy"></dd></dl></dd></dl>

<ul><li>The general p-th raw moment (p not necessarily an integer) can be expressed as<a href="#cite_note-4">[4]</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} \left[\left|N\left(\mu ,\sigma ^{2}\right)\right|^{p}\right]&={\frac {\left(2\sigma ^{2}\right)^{p/2}\Gamma \left({\tfrac {1+p}{2}}\right)}{\sqrt {\pi }}}\ {}_{1}F_{1}\left(-{\tfrac {p}{2}},{\tfrac {1}{2}},-{\tfrac {\mu ^{2}}{2\sigma ^{2}}}\right)\\\operatorname {E} \left[N\left(\mu ,\sigma ^{2}\right)^{p}\right]&=\left(-2\sigma ^{2}\right)^{p/2}U\left(-{\tfrac {p}{2}},{\tfrac {1}{2}},-{\tfrac {\mu ^{2}}{2\sigma ^{2}}}\right)\end{aligned}}}</annotation>

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</math><img src="./b1f3c2f689750465c3859d236366e75c3e40b20b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:56.759ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} \left[\left|N\left(\mu ,\sigma ^{2}\right)\right|^{p}\right]&={\frac {\left(2\sigma ^{2}\right)^{p/2}\Gamma \left({\tfrac {1+p}{2}}\right)}{\sqrt {\pi }}}\ {}_{1}F_{1}\left(-{\tfrac {p}{2}},{\tfrac {1}{2}},-{\tfrac {\mu ^{2}}{2\sigma ^{2}}}\right)\\\operatorname {E} \left[N\left(\mu ,\sigma ^{2}\right)^{p}\right]&=\left(-2\sigma ^{2}\right)^{p/2}U\left(-{\tfrac {p}{2}},{\tfrac {1}{2}},-{\tfrac {\mu ^{2}}{2\sigma ^{2}}}\right)\end{aligned}}}" loading="lazy"></dd></dl></dd>

<dd>In the second formula the function's second <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a> can be chosen by multiplying with (−1)p.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Application_to_continued_fractions">Application to continued fractions</h2></div>

By applying a limiting argument to <a href="Gauss's_continued_fraction" title="Gauss's continued fraction">Gauss's continued fraction</a> it can be shown that<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {M(a+1,b+1,z)}{M(a,b,z)}}={\cfrac {1}{1-{\cfrac {\displaystyle {\frac {b-a}{b(b+1)}}z}{1+{\cfrac {\displaystyle {\frac {a+1}{(b+1)(b+2)}}z}{1-{\cfrac {\displaystyle {\frac {b-a+1}{(b+2)(b+3)}}z}{1+{\cfrac {\displaystyle {\frac {a+2}{(b+3)(b+4)}}z}{1-\ddots }}}}}}}}}}}</annotation>

</semantics>

</math><img src="./5416c59087ba9b0050c31b9f5db280274e6e4dff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -30.171ex; width:57.311ex; height:34.343ex;" alt="{\displaystyle {\frac {M(a+1,b+1,z)}{M(a,b,z)}}={\cfrac {1}{1-{\cfrac {\displaystyle {\frac {b-a}{b(b+1)}}z}{1+{\cfrac {\displaystyle {\frac {a+1}{(b+1)(b+2)}}z}{1-{\cfrac {\displaystyle {\frac {b-a+1}{(b+2)(b+3)}}z}{1+{\cfrac {\displaystyle {\frac {a+2}{(b+3)(b+4)}}z}{1-\ddots }}}}}}}}}}}" loading="lazy"></dd></dl>

and that this continued fraction converges uniformly to a <a href="Meromorphic_function" title="Meromorphic function">meromorphic function</a> of z in every bounded domain that does not include a pole.

<ul><li><a href="Composite_B%C3%A9zier_curve" title="Composite Bézier curve">Composite Bézier curve</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFCampos2001" class="citation journal cs1">Campos, L.M.B.C. (2001). "On Some Solutions of the Extended Confluent Hypergeometric Differential Equation". Journal of Computational and Applied Mathematics. 137 (1): 177–200. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001JCoAM.137..177C">2001JCoAM.137..177C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0377-0427%2800%2900706-8">10.1016/s0377-0427(00)00706-8</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1865885">1865885</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFAndrewsAskeyRoy2001" class="citation book cs1">Andrews, G.E.; Askey, R.; Roy, R. (2001). Special functions. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0521789882</bdi>.</cite>.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> This is derived from Abramowitz and Stegun (see reference below), <a rel="nofollow" class="external text" href="http://people.math.sfu.ca/~cbm/aands/page_508.htm">page 508</a>, where a full asymptotic series is given. They switch the sign of the exponent in exp(iπa) in the right half-plane but this is immaterial, as the term is negligible there or else a is an integer and the sign doesn't matter.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.wiley.com/en-us/Aspects+of+Multivariate+Statistical+Theory-p-9780471769859">"Aspects of Multivariate Statistical Theory | Wiley"</a>. Wiley.com. Retrieved 2021-01-23.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFFrank1956" class="citation journal cs1">Frank, Evelyn (1956). "A new class of continued fraction expansions for the ratios of hypergeometric functions". Trans. Am. Math. Soc. 81 (2): 453–476. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1956-0076937-0">10.1090/S0002-9947-1956-0076937-0</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1992927">1992927</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0076937">0076937</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_504.htm">"Chapter 13"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 504. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite></li>

<li><cite id="CITEREFChistova2001" class="citation cs2">Chistova, E.A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Confluent_hypergeometric_function">"Confluent hypergeometric function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFDaalhuis2010" class="citation cs2">Daalhuis, Adri B. Olde (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/13">"Confluent hypergeometric function"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li>

<li><cite id="CITEREFErdélyiMagnusOberhettingerTricomi1953" class="citation book cs1"><a href="Arthur_Erd%C3%A9lyi" title="Arthur Erdélyi">Erdélyi, Arthur</a>; <a href="Wilhelm_Magnus" title="Wilhelm Magnus">Magnus, Wilhelm</a>; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). Higher transcendental functions. Vol. I. New York–Toronto–London: McGraw–Hill Book Company, Inc. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0058756">0058756</a>.</cite></li>

<li><cite id="CITEREFKummer1837" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Ernst_Eduard_Kummer" class="mw-redirect" title="Ernst Eduard Kummer">Kummer, Ernst Eduard</a> (1837). <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002141329">"De integralibus quibusdam definitis et seriebus infinitis"</a>. <a href="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a> (in Latin). 1837 (17): 228–242. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1837.17.228">10.1515/crll.1837.17.228</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121351583">121351583</a>.</cite></li>

<li><cite id="CITEREFSlater1960" class="citation book cs1"><a href="Lucy_Joan_Slater" title="Lucy Joan Slater">Slater, Lucy Joan</a> (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/confluenthyperge0000slat">Confluent hypergeometric functions</a>. Cambridge, UK: Cambridge University Press. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0107026">0107026</a>.</cite></li>

<li><cite id="CITEREFTricomi1947" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Francesco_Giacomo_Tricomi" class="mw-redirect" title="Francesco Giacomo Tricomi">Tricomi, Francesco G.</a> (1947). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02415375">"Sulle funzioni ipergeometriche confluenti"</a>. Annali di Matematica Pura ed Applicata. Series 4 (in Italian). 26: 141–175. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02415375">10.1007/bf02415375</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-4622">0003-4622</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0029451">0029451</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119860549">119860549</a>.</cite></li>

<li><cite id="CITEREFTricomi1954" class="citation book cs1 cs1-prop-foreign-lang-source">Tricomi, Francesco G. (1954). Funzioni ipergeometriche confluenti. Consiglio Nazionale Delle Ricerche Monografie Matematiche (in Italian). Vol. 1. Rome: Edizioni cremonese. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-88-7083-449-9</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0076936">0076936</a>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)</li>

<li><cite id="CITEREFOldhamMylandSpanier2010" class="citation book cs1">Oldham, K.B.; Myland, J.; Spanier, J. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UrSnNeJW10YC&pg=PA75">An Atlas of Functions: with Equator, the Atlas Function Calculator</a>. Springer New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-48807-3</bdi>. Retrieved 2017-08-23.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://dlmf.nist.gov/13">Confluent Hypergeometric Functions</a> in NIST Digital Library of Mathematical Functions</li>

<li><a rel="nofollow" class="external text" href="http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F1/">Kummer hypergeometric function</a> on the Wolfram Functions site</li>

<li><a rel="nofollow" class="external text" href="http://functions.wolfram.com/HypergeometricFunctions/HypergeometricU/">Tricomi hypergeometric function</a> on the Wolfram Functions site</li></ul>

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